Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/56

16 to indicate the direction, as well as the magnitude, of the most rapid decrease of $$\Psi$$.

18.] There are cases, however, in which the conditions $$ \frac{dZ}{dy}-\frac{dY}{dz}=0,\quad \frac{dX}{dz}-\frac{dZ}{dx}=0,\quad$$ and $$\quad\frac{dY}{dx}-\frac{dX}{dy}=0, $$ which are those of $$Xdx + Ydy + Zdz$$ being a complete differential, are fulfilled throughout a certain region of space, and yet the line integral from $$A$$ to $$P$$ may be different for two lines, each of which lies wholly within that region. This may be the case if the region is in the form of a ring, and if the two lines from $$A$$ to $$P$$ pass through opposite segments of the ring. In this case, the one path cannot be transformed into the other by continuous motion without passing out of the region.

We are here led to considerations belonging to the Geometry of Position, a subject which, though its importance was pointed out by Leibnitz and illustrated by Gauss, has been little studied. The most complete treatment of this subject has been given by J. B. Listing.

Let there be $$p$$ points in space, and let $$l$$ lines of any form be drawn joining these points so that no two lines intersect each other, and no point is left isolated. We shall call a figure composed of lines in this way a Diagram. Of these lines, $$p-1$$ are sufficient to join the $$p$$ points so as to form a connected system. Every new line completes a loop or closed path, or, as we shall call it, a Cycle. The number of independent cycles in the diagram is therefore $$k=l-p+1$$.

Any closed path drawn along the lines of the diagram is composed of these independent cycles, each being taken any number of times and in either direction. The existence of cycles is called Cyclosis, and the number of cycles in a diagram is called its Cyclomatic number.

Surfaces are either complete or bounded. Complete surfaces are either infinite or closed. Bounded surfaces are limited by one or more closed lines, which may in the limiting cases become finite lines or points.

A finite region of space is bounded by one or more closed surfaces. Of these one is the external surface, the others are